Design and analysis of a computer experiment for the study of the distortion of an advanced gas-cooled reactor moderator brick

Highlights

• A computer experiment for studying the deformation of an AGR moderator brick was run.

• A computationally efficient methodology was used for assessing parameter sensitivity.

• A subset of shape metrics for use in future parameter calibration was proposed.

Abstract

Engineering models, resolved using the finite element method, are used to model through-life changes in material properties and the resulting internal brick stresses in graphite moderator bricks within an advanced gas-cooled reactor (AGR). These models require inputs that describe the loading conditions of the moderator brick and coded relationships describing the behaviour of material properties as the brick ages. A highly multivariate spacio-temporal dataset of displacements, stresses and strains is returned as model output. In this work we describe a computer experiment conducted to study the variability in brick distortion resulting from parameter value uncertainty in the inputs to the model. We identify summary measures of brick shape and conduct global sensitivity analyses to identify the key uncertain parameters driving the variability in brick shape. Our results indicate that different measures of brick shape show sensitivities to different groups of model inputs and suggest that data obtained from core monitoring campaigns can be used to calibrate uncertain parameters in the model. Model calibration and the impact of calibration on predictions of internal brick stresses, which cannot be directly validated, are areas of current research.

Introduction

The advanced gas-cooled reactor (AGR) is a second-generation nuclear reactor which is constructed from a large number of hollow cylindrical graphite bricks. These bricks are building blocks of the core and form channels for fuel, control rods, and coolant (Steer, 2007). The graphite also acts as a neutron moderator. The AGRs are carbon dioxide cooled although the coolant also contains additives; principal amongst these is methane (Nonbel, 1996) at a concentration of approximately 200 parts per million of the circulating gas. The use of carbon dioxide, under approximately 40 bar pressure, as a coolant in AGRs, avoids the possibility of phase change during fault conditions (Tsang and Marsden, 2006).

Over the lifetime of the AGR there is a significant degradation of the graphite components, quantified through changes to material properties of the graphite compared with the start-of-life (un-irradiated) condition. Fast neutron damage and the high operating temperatures within the reactor core are the primary causes of this degradation (Fahad et al., 2013, Fahad et al., 2015). Radiolytic oxidation of the graphite, which is quantified as a loss of mass or weight loss relative to the start-of-life condition (Brocklehurst et al., 1970) is a further damage mechanism. Information from Materials Test Reactor (MTR) programmes have provided important data for quantifying changes to the graphite compared with its start-of-life condition (Eason et al., 2013a, Eason et al., 2013b, Eason et al., 2013c). Although the graphite core is a fairly inaccessible structure, during statutory and refuelling outages of a reactor, visual inspection of fuel channel bores is possible and dimensional measurements of moderator brick bores are taken. Furthermore, the removal of small specimens of irradiated graphite from a small subset of moderator bricks during statutory outages allows material properties measurements to be made. Thus, a longitudinal dataset on various material properties has been collated over several decades of reactor operation. Bore measurement and material properties data obtained during inspection campaigns at operating reactors (as described above) augments the data obtained from MTR experiments.

Core monitoring is supplemented by engineering models of graphite components, which are required in order to study important processes in-reactor that cannot be directly observed, including coolant flows, control rod entry margins, key/keyway clearances and graphite weight loss. An engineering model of the moderator brick, resolved using the finite element method (Tsang and Marsden, 2006, Jones, 2007) is used to study internal brick stresses that evolve due to changes to material properties of the graphite and the loading conditions in-reactor; from predictions of stresses, the likelihood of bricks subsequently developing cracks may be predicted. As input these models take through-life loading conditions quantified by field variables for weight loss, fast neutron irradiation and temperature and materials property models that quantify changes to the microstructure of the graphite (as a function of temperature, fast neutron irradiation and weight loss). A highly multivariate output of spatially and longitudinally varying displacements, strains and stresses is produced as model output.

However, engineering models failed to predict early-life bore-initiated cracking observed during core inspection in a subset of moderator bricks (Maul et al., 2011, Tan et al., 2013), and there is uncertainty in the onset timing and rate of later-life cracking initiated from the keyway-root of moderator bricks (McNally et al., 2016). Recent computer experiments (McNally et al., 2016, Tan et al., 2017) have investigated how perturbations to uncertain parameters in affect predictions of strength and internal brick stresses. Through tuning uncertainty parameters in engineering models such that the model output (predictions of stresses and strength) is consistent with the pattern of cracking observed to date, more confidence in the predictive capability of these models might be gained. This approach can be viewed as indirect calibration since neither internal brick stresses nor strengths can be observed – the calibration is to an observable outcome and, even if the model is consistent with such observations, it only weakly validates the model predictions. Furthermore, the appropriate failure criterion to apply is in itself uncertain.

The isotropy of nuclear grade graphite (i.e. the anisotropy factor) can be defined by the ratio of the CTEs measured in the two main crystallographic directions (Haag et al., 1990). Gilsocarbon has an isotropy factor of 1.05 (Preston and Marsden, 2006) and hence, is often referred to as being near-isotropic. This leads to different magnitudes of dimensional changes in the axial (against-grain), radial (with-grain), and circumferential (with-grain) directions with respect to the brick axes (Fahad et al., 2017). The irradiation-induced dimensional changes of the graphite lead to identifiable changes in the shape of the moderator bricks and exemplifications are given in Fig. 1. Distortion at the brick bore is illustrated in Fig. 2 for four time points: 0, 2, 10 and 30 full power years (fpy) corresponding to start-of-life, low burn-up, early-life and late-life states. The primary change in shape is a barrelling of the brick due to the brick shrinking at a faster rate at the bore compared with the periphery (due to temperature, fast neutron irradiation and weight loss gradients and resultant non-uniform loading through the thickness of the brick (Fig. 2, 2fpy), with a smaller change occurring due to a non-uniform dose profile through the height of the brick resulting from small gaps between fuel elements (McLachlan et al., 1995, Moskovic, 2014) (Fig. 2 10fpy). A wheatsheaf shape occurs in later life as a consequence of the periphery of the brick shrinking at a greater rate than the brick bore (Fig. 2, 30 fpy).

These shape changes, quantified as displacements of nodes in the FE model compared with their start-of-life position, are returned as model output, and can be compared against measured bore shapes obtained during channel inspections. McNally et al. (2014) adapted an existing model for dimensional change (Eason et al, 2006) in an inert environment such that it was appropriate for the oxidising environment of the AGR and subsequently developed a novel approach for calibrating three parameters in the engineering model relating to the with-grain dimensional change material property relationship by tuning model parameters such that predictions of diameter at mid-height of the moderator brick were consistent with measured diameters in moderator bricks. Recent work (Fahad et al., 2017) has investigated whether the modest ovalisation of the moderator brick bore, compared with the cylindrical profile at start-of-life, could be informative about the irradiation creep material property relationship.

In this work we examine a wider dataset of moderator brick shape changes obtained from the computer experiment described in Fahad et al. (2017). The objective of the novel work described in the paper was to identify measures of brick shape that varied substantially over the model runs from the computer experiment, and where the feature could also be reliably measured in core monitoring data. This can be viewed as preparatory work preceding attempts at parameter calibration.

The remainder of the paper is set-out as follows. In Section 2 we provide brief details about the modelled components and an overview of the FE modelling approach used in this work. In Section 3 we describe the information available from core monitoring and identify measures of brick deformation for further study. In Section 4 we describe the computer experiment, make some comparisons of the spread of model predictions against core monitoring data, and undertake a global sensitivity using the computationally efficient approach of Oakley and O’Hagan (2004) and a second qualitative method of analysis based on presence or absence of brick features. Finally, we discuss the relationship between brick shape and internal brick stresses and the utility of using brick shape measurements to improve knowledge about model outputs (stresses and strains) that cannot be directly validated.